From: Stuart Armstrong (email@example.com)
Date: Sat Mar 22 2008 - 03:43:45 MDT
Some extra precisions on my long lost example (long lost, since it
happened two long weeks ago...)
> > The answer, whatever it is, will have simpler, quasi answers -
> > incomplete but informative. The hash function equivalents
> > will not be simpler than the hash function equivalent to the full
> > question.
Examples of simpler questions: for the derivative, we have interim
results: the derivative of a monomial is a monomial, the derivative of
a polynomial of n-th degree is a polynomial of n-1 th degree. These
are simpler than knowing the full derivative rule, but to list them
under the hash function is actualy more complicated than listing the
GLUT for the derivative itself (as we first have to define the subsets
"monomials" and "polnomial of n-th degree").
> > 2) The implicit infinity.
> > Implicit in the definition of differentiation is the fact that we
> > could differentiate any polynomial (with, say, rational coefficients).
> > The definition of differentiation to f(S) does not extend to infinity
> > in this way; in fact, there is no evident extention of f(S).
> What? Even if you have infinitely many polynomials, each has a
> derivative, the only difference being that there is no "top dog".
> Strategically, would it have been better to stick with a crippled
> definition of derivative, oh, something along the lines of either
> "we are not going to consider S to be infinite", or "a qDerivative
> is a derivative of a polynomial function, but the domain of
> qDerivative is by definition finite"? Or something like that?
The derivative can be defined for any polynomial, and can be defined
in a finite (and very short) sentence. A finite definition, generating
an infinite number of relations.
Now try and go the other way; from a GLUT, try and deduce the general
rule. If we have the GLUT in polynomial form, the universal rule is
easy to deduce; if we have the GLUT in some hash function equivalent,
we can't do so.
> I think that the whole infinity digression is a red-herring to your otherwise
> brilliant analysis (and I don't often call what other posters do "brilliant"),
> reserving that appellation strictly to apply to my own missives only.
I'm actually blushing there :-) Many thanks. However, the red herring
is still, in my view, a distinction between a GLUT and a
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